In [1]}, the authors defined a Donaldson-Thomas motive, gluing the motive of vanishing cycles defined using arc spaces. Motives glue in the Zariski topology and not in the étale topology, hence to prove a localization formula one must a priori restrict to d-critical schemes admitting Zariski \(\mathbb {C}^\ast \) -equivariant critical charts. According to [2]}, the existence of such charts is equivalent to the existence of a \(\mathbb {C}^\ast \) -equivariant Zariski cover (such a \(\mathbb {C}^\ast \) -action is then called 'good'), which holds for normal spaces from Sumihiro's theorem [3]}, but not in greater generality (see example 2.46 of [2]}). Davesh Maulik has proven a formula similar to (REF ) for motivic DT invariants on d-critical schemes with a good circle-compact \(\mathbb {C}^\ast \) -action, as explained in section 5.3 of [5]}, in an unpublished preprint (private communication). A generalization of this result for non-Archimedean geometry was proven after in [6]}. For non circle-compact actions one can compute the DT motive of the attracting variety, which is by definition circle-compact, by this way.
